Preheat grill or broiler to medium heat. Place mushrooms gill-side up on a work surface and sprinkle with 2 tablespoons olive oil, balsamic vinegar, a little kosher salt (optional), and remaining 1 teaspoon pepper. With remaining 1/2 tablespoon oil, lightly coat the rounded side of the mushroom caps.

Place mushrooms, gill-side down, on grill or broiler pan. Grill or broil for 6–8 minutes on each side, until soft and slightly charred.

Transfer mushrooms to a roasting pan and fill with 10–15 tomatoes each. Top with cheese. Place in hot oven until cheese bubbles. Set each mushroom on a mound of steamed spinach and serve immediately.